Subgroups — C₆₇×#22253 #30701504

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2331	C₂	1	2	yes	maximal
3	C₆₇×#1993	C₂	1	2	yes	maximal
4	C₆₇×#1920	C₂	1	4	yes	maximal
5	C₆₇×#1919	C₂	1	4	yes	maximal
6	C₆₇×#1835	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
9	#498×C₆₇	C₄	1	8	yes	complemented
10	C₂×#106×C₆₇	C₂²	1	4	yes	—
11	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#391×C₆₇	—	2	4	no	—
14	#389×C₆₇	C₂²	1	4	yes	—
15	#387×C₆₇	—	2	4	no	—
16	#382×C₆₇	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
18	#376×C₆₇	C₂²	1	1	yes	characteristic
19	#340×C₆₇	C₂²	1	4	yes	—
20	#340×C₆₇	C₂²	1	4	yes	—
21	#340×C₆₇	C₂²	1	4	yes	—
22	#336×C₆₇	C₂²	1	2	yes	—
23	#336×C₆₇	C₂²	1	2	yes	—
24	C₂²×Q₈×C₆₇	—	2	2	no	—
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
28	C₄⋊Q₈×C₆₇	—	2	4	no	—
29	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
34	C₂×C₄²×C₆₇	—	2	4	no	—
35	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	#106×C₆₇	C₂×C₄	1	16	yes	—
38	#97×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
40	#95×C₆₇	—	2	4	no	—
41	#95×C₆₇	—	2	2	no	—
42	#95×C₆₇	—	2	2	no	—
43	#95×C₆₇	—	2	4	no	—
44	#95×C₆₇	D₄	1	4	yes	—
45	C₂×Q₈×C₆₇	—	2	8	no	—
46	C₂×Q₈×C₆₇	—	8	4	no	—
47	C₂²×C₄×C₆₇	—	2	4	no	—
48	C₂²×C₄×C₆₇	—	2	4	no	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
59	C₄⋊C₄×C₆₇	—	2	8	no	—
60	C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₄⋊C₄×C₆₇	—	8	4	no	—
64	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	—	8	4	no	—
68	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₆₇	#41	1	2	yes	—
70	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
71	Q₈×C₆₇	—	8	8	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	8	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	—	4	4	no	—
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
86	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
87	C₈×C₆₇	—	2	4	no	—
88	C₈×C₆₇	—	4	1	no	—
89	C₈×C₆₇	—	4	1	no	—
90	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
91	C₂²×C₆₇	#536	1	2	yes	—
92	C₂²×C₆₇	#577	1	4	yes	—
93	C₄×C₆₇	—	2	4	no	—
94	C₄×C₆₇	—	4	8	no	—
95	C₄×C₆₇	—	4	4	no	—
96	C₄×C₆₇	—	8	4	no	—
97	C₄×C₆₇	#519	1	2	yes	—
98	C₄×C₆₇	#541	1	1	yes	characteristic
99	C₄×C₆₇	#541	1	1	yes	characteristic
100	#22253	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂×C₆₇	#2327	1	1	yes	characteristic
102	C₂×C₆₇	#2949	1	2	yes	—
103	C₂×C₆₇	#3244	1	4	yes	—
104	C₂×#498	C₂×C₆₇	1	1	yes	characteristic
105	#2331	C₂×C₆₇	1	2	yes	—
106	#1993	C₂×C₆₇	1	2	yes	—
107	#1920	C₂×C₆₇	1	4	yes	—
108	#1919	C₂×C₆₇	1	4	yes	—
109	#1835	C₂×C₆₇	1	1	yes	characteristic
110	#1746	C₂×C₆₇	1	1	yes	characteristic
111	C₆₇	#22253	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
113	#498	C₄×C₆₇	1	8	yes	complemented
114	C₂×#106	C₂²×C₆₇	1	4	yes	—
115	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
117	#391	—	2	4	no	—
118	#389	C₂²×C₆₇	1	4	yes	—
119	#387	—	2	4	no	—
120	#382	C₂²×C₆₇	1	2	yes	—
121	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
122	#376	C₂²×C₆₇	1	1	yes	characteristic
123	#340	C₂²×C₆₇	1	4	yes	—
124	#340	C₂²×C₆₇	1	4	yes	—
125	#340	C₂²×C₆₇	1	4	yes	—
126	#336	C₂²×C₆₇	1	2	yes	—
127	#336	C₂²×C₆₇	1	2	yes	—
128	C₂²×Q₈	—	2	2	no	—
129	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
132	C₄⋊Q₈	—	2	4	no	—
133	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
137	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
138	C₂×C₄²	—	2	4	no	—
139	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
140	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
141	#106	C₂×C₄×C₆₇	1	16	yes	—
142	#97	—	2	2	no	—
143	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
144	#95	—	2	4	no	—
145	#95	—	2	2	no	—
146	#95	—	2	2	no	—
147	#95	—	2	4	no	—
148	#95	D₄×C₆₇	1	4	yes	—
149	C₂×Q₈	—	2	8	no	—
150	C₂×Q₈	—	8	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
154	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
155	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	8	4	no	—
168	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
169	C₄²	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	8	4	no	—
172	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
173	C₄²	#41×C₆₇	1	2	yes	—
174	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
175	Q₈	—	8	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
187	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
188	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
189	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
190	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	4	1	no	—
193	C₈	—	4	1	no	—
194	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
195	C₂²	#536×C₆₇	1	2	yes	—
196	C₂²	#577×C₆₇	1	4	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	4	8	no	—
199	C₄	—	4	4	no	—
200	C₄	—	8	4	no	—
201	C₄	#519×C₆₇	1	2	yes	—
202	C₄	#541×C₆₇	1	1	yes	characteristic
203	C₄	#541×C₆₇	1	1	yes	characteristic
204	C₂	C₆₇×#2327	1	1	yes	characteristic
205	C₂	C₆₇×#2949	1	2	yes	—
206	C₂	C₆₇×#3244	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
9. #498×C₆₇	C₄	semidirect
100. #22253	C₆₇	direct (N × H)
111. C₆₇	#22253	direct (N × H)
113. #498	C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#498×C₆₇	2
C₆₇×#2331	2
C₆₇×#1993	2
C₆₇×#1920	2
C₆₇×#1919	2
C₆₇×#1835	2
C₆₇×#1746	2
#22253	67

Subgroups — C67×#22253 #30701504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22253 #30701504